Executive Summary : | This project aims to investigate reality problems for linear Lie groups and classify the "infinitesimal reality" that comes from the adjoint action of a Lie group. A linear Lie group acts on its Lie algebra $\g$ by conjugation, and an element $X\in \g$ is called ${\rm Ad}_G$-real if $-X =gXg^-1} $ for some $g\in G$. An ${\rm Ad}_G$-real element $X$ is called strongly ${\rm Ad}_G$-real if $-X = \tau X \tau^-1} $ for some involution $\tau\in G$. The problem can be restated as the problem to classify the'real' and'strongly real' adjoint orbits of semisimple Lie groups. One such group is the classical M\"obius group ${PSL}(2, \C)$ of the two-dimensional sphere that acts as the orientation-preserving isometry group of the three-dimensional hyperbolic space. This group is of interest in three-dimensional topology due to its connection with discrete subgroups and geometric structures arising from them. Basmajian and Maskit have investigated decomoposability properties in $n$-dimensional M\"obius groups, but there has not been a complete classification of such elements. The project aims to investigate this problem for other families of Lie groups, including the isometry group ${Sp}(2,1)$ of the quaternionic hyperbolic space and the special linear groups ${\rm SL}(n, K)$. The expected outcome is a complete understanding of reversibility or reality in linear Lie groups and their geometric applications, such as decomposability results. These investigations may also impact understanding of the geometric structures that arise from Lie groups. |